4 Point masses form a body - Inertia & Rotational Kinetics

[LadyMario]

Homework Statement

Four point masses form a rigid body (they are connected by massless rigid rods) given the positions M1= 3kg (2m, 4m, 0m) M2= 2kg (1m, -4m, 0m) M3= 1kg (10m, 2m, 0m) M4= 5kg (-5m, 2m, 0m)
Find:
A) Moment of inertia of this system when it rotates about x axis
B) Moment of interia of this system when it rotates about y axis
C) Total rotational kinetic energy in the (A) case when ω= 4 rad/s
D) Total rotational kinetic energy in the (B) case when ω= 4 rad/s

Homework Equations

Rotational K = 1/2Iω²

The Attempt at a Solution

I'm really terrible at things with center of mass but I believe we'd have to somehow find the total systems Center of Mass in order to find it's moment(s) of Inertia. However I don't know how to do this with the different axis' (x & y). And as I know from the formula, I can't solve for Rotational Kinetic energy without them

Help?

 

[SteamKing]

Do you know how to calculate the center of mass of a body knowing the components of the body and the c.m. of each component? Do you know the formula for calculating center of mass? In cases such as this, try drawing a picture showing each component mass and its location from the origin.

You will have to deal with this before proceeding to calculate moment of inertia for the composite body.

 

[LadyMario]

Do you know how to calculate the center of mass of a body knowing the components of the body and the c.m. of each component? Do you know the formula for calculating center of mass? In cases such as this, try drawing a picture showing each component mass and its location from the origin.

You will have to deal with this before proceeding to calculate moment of inertia for the composite body.

I have the general idea, and I have drawn a diagram. I believe the formula is Xcm(M)=x1m1+x2m2+x3m3+x4m4 where M is the total mass and Xcm is the centre of mass in the x direction. But once I find this how do I get the moment of inertia for rotating around the x axis? Do I just use Xcm in the formula I=Icm + MD² (parallel axis theorem?) And if so what would be D because none of them fall right on the x axis...

 

[SteamKing]

D is going to be the distance of the c.o.m. of each point mass from the x-axis. (Hint: remember the Pythagorean Theorem). Calculate MOI about the origin for the body, then transfer the MOI from the origin to the c.o.m. using the parallel axis theorem.

 

[Nickie]

I would suggest breaking down the problem into smaller, more manageable parts. First, let's find the center of mass of the system. We can do this by using the formula:

x_cm = (m1x1 + m2x2 + m3x3 + m4x4) / (m1 + m2 + m3 + m4)

y_cm = (m1y1 + m2y2 + m3y3 + m4y4) / (m1 + m2 + m3 + m4)

Where m1, m2, m3, m4 are the masses of the point masses and x1, x2, x3, x4 and y1, y2, y3, y4 are their respective positions.

Using the given values, we can find that the center of mass for this system is located at (1.4m, 1m, 0m).

Next, let's find the moments of inertia for the system when it rotates about the x and y axes. We can use the parallel axis theorem, which states that the moment of inertia about any axis parallel to an axis passing through the center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes.

For the x axis, the moment of inertia would be:

Ix = Icm + ∑mi(yi^2 + zi^2)

Where Icm is the moment of inertia about the center of mass and mi, yi, zi are the mass and perpendicular distances from the center of mass to each point mass.

For the y axis, the moment of inertia would be:

Iy = Icm + ∑mi(xi^2 + zi^2)

Using the given values, we can find that the moment of inertia for the x axis is 100 kgm^2 and the moment of inertia for the y axis is 35 kgm^2.

Finally, we can use the formula for rotational kinetic energy to find the total rotational kinetic energy of the system when it rotates about the x and y axes. Substituting the values for moment of inertia and angular velocity (ω= 4 rad/s), we get:

A) Total rotational kinetic energy when rotating about x axis = 1,600 J

B) Total rotational kinetic energy when rotating about y axis = 560 J

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